Energy-preserving Runge-kutta Methods
نویسندگان
چکیده
We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems. Mathematics Subject Classification. 65P10, 65L06. Received May 21, 2008. Published online July 8, 2009. All Runge-Kutta (RK) methods preserve arbitrary linear invariants [12], and some (the symplectic) RK methods preserve arbitrary quadratic invariants [4]. However, no RK method can preserve arbitrary polynomial invariants of degree 3 or higher of arbitrary vector fields [1]; the linear system ẋ = x, ẏ = y, ż = −2z, with invariant xyz, provides an example. (Preservation would require R(h)2R(−2h) ≡ 1, where R is the stability function of the method; but this requires R(h) = e, which is impossible [6].) This result does not rule out, however, the existence of RK methods that preserve particular (as opposed to arbitrary) invariants. Since the invariant does not appear in the RK method, this will require some special relationship between the invariant and the vector field. Such a relationship does exist in the case of the energy invariant of canonical Hamiltonian systems: see [3,5] on energy-preserving B-series. In this article we show that for any polynomial Hamiltonian function, there exists an RK method of any order that preserves it. The key is the average vector field (AVF) method first written down in [9] and identified as energy-preserving and as a B-series method in [10]: for the differential equation ẋ = f(x), x ∈ R, (1) the AVF method is the map x → x′ defined by x′ − x h = ∫ 1 0 f(ξx′ + (1− ξ)x) dξ. (2)
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